Many electronic circuits that are known to the art rely upon electrical resistors as building blocks of more complex circuits. Some circuits require resistors with comparatively large resistance values. In simple discrete circuits such as those commonly found on printed circuit boards (PCBs) discrete resistor components that are well-known to the art provide resistance values for a wide range of circuits. However, many modern circuit implementations rely on integrated circuits that use, for example, a silicon wafer as a substrate with various circuit components etched into the silicon wafer and connected together via metal interconnect layers using processes that are known to the art. In these circuits, the physical size of circuit elements including resistors directly affects the size of the circuit where those of skill in the art realize that one goal of modern circuit design is to produce smaller integrated circuits that can be manufactured more economically. One disadvantage of producing a resistor with a high resistance value in an integrated circuit is that a large resistor typically occupies a larger physical area of the circuit, which increases the overall size of the circuit.
One prior-art solution produces a resistor with a comparatively small physical size and a large resistance using a switched resistor configuration. FIG. 1 depicts an RC circuit 100 that incorporates a prior switched resistor 102 that further includes a resistor 104 (R) and a switch 108. The RC circuit also includes a voltage source 128 and a load capacitor 132. The circuit 100 of FIG. 1 depicts a simple RC filter but the switched resistor 102 can be incorporated in a wide range of other circuits that employ a resistor. The circuit 100 of FIG. 1 depicts an idealized configuration that omits the effects of the inherent parasitic capacitance Cp of the resistor R for explanatory purposes, although the effects of the parasitic capacitance are described below. The prior-art circuit schematic 140 is a distributed resistance model of the switched resistor 102 that includes a series connection of a single switch 108 and multiple resistors 104A-104N that each include a parasitic capacitance Cp 110A-110N. In the embodiment of FIG. 1, the resistors 104A-104N are each smaller than the resistor 104 and the sum of the resistances 104A-104N and the parasitic capacitances 110A-110N is equal to the resistance of the resistor 104 and the parasitic capacitance 110, respectively. The switched resistor model 140 produces the same effective resistance as the switched resistor 102 and is affected by parasitic capacitance in substantially the same manner as the switched resistor 102.
During operation, a clock source (not shown) operates the switch ϕ1 operates at a predetermined frequency and duty cycle to close the switch ϕ1 only during the period Ton for each clock cycle Tp. When the switch 108 is opened the resistor 104 presents effectively infinite resistance and when the switch ϕ1 is closed during Ton the resistor R presents the inherent resistance R of the resistor to the voltage source. The ideal effective resistance of the resistors 104 or 104A-104N that ignores the effects of parasitic capacitance becomes
      R    effi    =      R          (                        T          on                          T          p                    )      where the ratio of Ton and Tp is also referred to as the duty cycle D
            (              D        =                  (                                    T              on                                      T              p                                )                    )        ⁢                  ⁢    and    ⁢                  ⁢          R      effi        =            R      D        .  While the precise period of the clock cycle Tp varies between embodiments, some prior-art switched resistors operate with a clock cycle in the kilohertz range (e.g. 25 KHz with a clock period time of Tp=4×10−5 sec) and with duty cycles, on the order of 3.13×10−2 that close the switch ϕ1 during the period (Ton) for a pulse time of 1.25×10−6 sec (1.25 μsec) per clock cycle. Thus, the switched resistor circuit effectively produces a much larger average resistance value than the inherent resistance of the resistor R, which enables integrated circuit embodiments to use a resistor that occupies a comparatively small amount of space in the integrated circuit. In the embodiment of the simple RC filter in FIG. 1, the increased resistance of the switched resistance device enables the filter to operate with a lower corner frequency (f3 dB), which is described as the 3 decibel (dB) cutoff frequency of the filter:
      f          3      ⁢                          ⁢      dB        =            1                        R          eff                ⁢                  C          L                      =                  D                  RC          L                    .      
The ideal resistance effective resistance Reffi described above omits the effects of the parasitic capacitance Cp. The parasitic capacitance Cp reduces the effective R in the switched resistance device of FIG. 1. In FIG. 1, the circuit 150 depicts a second, parallel resistance Rp that models the effects of the parasitic capacitance Cp when operating the switched resistor using a predetermined switching time period Tp. While switched resistance devices enable the use of a resistor with a smaller inherent resistance value to provide a larger effective resistance value, one problem that affects the prior art switched resistance device is that the parasitic capacitance that is inherent to the resistor tends to limit the maximum effective resistance that the switched resistance device produces in a practical circuit. In the ideal example that omits the parasitic capacitance the resistance
      R    effi    =      R    D  scales to large numbers simply by reducing the duty cycle D towards zero, but the parasitic capacitance Cp in the actual implementation of the circuit reduces the practical maximum resistance level. The equivalent circuit 150 in FIG. 1 is another model of the resistor R that depicts the effects of the parasitic capacitance as a parallel resistance with a value of
            T      p              C      p        .Thus, the total resistance Reff_pa for the prior-art switched resistor that incorporates the effects of the parasitic capacitance Cp yields the lower effective resistance value:
      R          eff      ⁢      _      ⁢      pa        =                    R        D            ❘              ❘                              T            p                                C            p                                =          R              D        +                              RC            p                                T            p                              where the “∥” notation indicates the two parallel resistances in the schematic diagram 150.
As set forth above, the effects of parasitic capacitance reduce the total effective resistance of the prior-art switched resistor 102. Additionally, the negative effects of the parasitic capacitance greatly increase in situations where the operating frequency of the switch increases and the corresponding time period Tp of each clock cycle in the switching signal decreases. For example, instead of the lower frequency of 25 KHz described above, many audio applications require that the switch operate a higher frequency of, for example, 50 KHz. For example, a prior art switched resistor with an inherent resistance of approximately 1.6×106Ω (1.6 MΩ) and a parasitic capacitance of approximately 7.91×10−13 F produces a total effective resistance of
      R                  eff        ⁢        _        ⁢        pa            ⁢                          ⁢              (                  25          ⁢                                          ⁢          KHz                )              =            R              D        +                              RC            p                                T            p                                =                            1.6          ×                      10            6                    ⁢                                          ⁢          Ω                                      3.13            ×                          10                              -                2                                              +                                                    (                                  1.6                  ×                                      10                    6                                    ⁢                                                                          ⁢                  Ω                                )                            ⁢                              (                                  7.91                  ×                                      10                                          -                      13                                                        ⁢                  F                                )                                                    4              ×                              10                                  -                  5                                            ⁢                                                          ⁢              sec                                          ≈              25.4        ×                  10          6                ⁢                                  ⁢        Ω            when using the 25 KHz clock signal (Tp=4×10−5 sec) and the duty cycle D=3.13×10−2. However, raising the clock signal to 50 KHz (Tp=2×10−5 sec) while holding all other parameters in the circuit equal produces a significantly lower effective resistance:
      R                  eff        ⁢        _        ⁢        pa            ⁢                          ⁢              (                  50          ⁢                                          ⁢          KHz                )              =            R              D        +                              RC            p                                T            p                                =                            1.6          ×                      10            6                    ⁢                                          ⁢          Ω                                      3.13            ×                          10                              -                2                                              +                                                    (                                  1.6                  ×                                      10                    6                                    ⁢                                                                          ⁢                  Ω                                )                            ⁢                              (                                  7.91                  ×                                      10                                          -                      13                                                        ⁢                  F                                )                                                    4              ×                              10                                  -                  5                                            ⁢                                                          ⁢              sec                                          ≈              16.9        ×                  10          6                ⁢                                  ⁢                  Ω          .                    
Thus, the increase in frequency produces a noticeable drop in the effective resistance of the prior art switched resistor since the time period Tp drops while the parasitic capacitance remains constant.
As depicted above, the parasitic capacitance reduces the effective resistance of the prior-art switched resistance device. Consequently, improvements to resistance devices that produce large resistances while reducing the negative effects of parasitic capacitance would be beneficial.